Due to their high degree of expressiveness, neural networks have recently been used as
surrogate models for map** inputs of an engineering system to outputs of interest. Once …

Traditional low-rank approximation is a powerful tool for compressing large data matrices
that arise in simulations of partial differential equations (PDEs), but suffers from high …

In the current paper, an efficient surrogate model based on combination of Proper
Orthogonal Decomposition (POD) and compressed sensing is developed for affordable …

The presence of uncertainty in material properties and geometry of a structure is ubiquitous.
The design of robust engineering structures, therefore, needs to incorporate uncertainty in …

We present a new convolution layer for deep learning architectures which we call
QuadConv—an approximation to continuous convolution via quadrature. Our operator is …

Mathematical modeling and simulation of complex physical systems based on partial
differential equations (PDEs) have been widely used in engineering and industrial …

In this paper, we study the multiscale Boltzmann equation with multi-dimensional random
parameters by a bi-fidelity stochastic collocation (SC) method developed in [52],[70],[71]. By …

Designing re-entry vehicles requires accurate predictions of hypersonic flow around their
geometry. Rapid prediction of such flows can revolutionize vehicle design, particularly for …

A ubiquitous challenge in design space exploration or uncertainty quantification of complex
engineering problems is the minimization of computational cost. A useful tool to ease the …

This thesis surveys and creates methods to allow for a mathematically consistent treatment
of non-uniform data sources in machine learning and data compression. These methods are …